Even though the book does not tell us why Atticus did not talk about his ability to shoot well (One-shot Finch), we can make an educated guess.
First, it has been a very long time since he shot a gun. For example, the children have never seen him shoot a gun. Therefore, what good is it to bring up something from the past. Atticus admits this to Heck Tate: “I haven’t shot a gun in thirty years—”
Second, Atticus is a man of humility. There is never a time in the book where there is even a hint of boasting. In fact, in everything he does, he is understated. Therefore, to tell people, even his children, that he was a dead-shot would be to be out of character.
Third, Atticus's personality is not one that fits with a rifle. He is a man of peace. Anything to do with violence he opposes, at least, in spirit. For example, when he went to protect Tom Robinson in jail and the mob came, Atticus had no weapon. It did not even cross his mind.